# Absolute Value in Algebra

### Absolute Value means ...

... **how far** a number is from zero:

"6" is 6 away from zero,

and "−6" is **also** 6 away from zero.

So the absolute value of 6 is **6**,

and the absolute value of −6 is also **6**

### Absolute Value Symbol

To show we want the absolute value we put "|" marks either side (called "bars"), like these examples:

|−5| = 5 | |7| = 7 |

The "|" can be found just above the enter key on most keyboards. |

## More Formal

More formally we have:

Which says the absolute value of x equals:

- x when x is greater than zero
- 0 when x equals 0
- −x when x is less than zero (this "flips" the number back to positive)

So when a number is positive or zero we leave it alone, when it is negative we change it to positive using −x.

### Example: what is |−17| ?

Well, it is less than zero, so we need to calculate "−x":

− ( −17 ) = +**17**

(Because two minuses make a plus)

## Useful Properties

Here are some properties of absolute values that can be useful:

- |a| ≥ 0 always!
That makes sense ... |a| can never be less than zero.

- |a| = √(a
^{2})Squaring a makes it positive or zero (for a as a Real Number). Then taking the square root will "undo" the squaring, but leave it positive or zero.

- |a × b| = |a| × |b|
Means these are the same:

- the absolute value of (a times b), and
- (the absolute value of a) times (the absolute value of b)

Which can also be useful when solving

- |u| = a is the same as u = ±a and vice versa
Which is often the key to solving most absolute value questions.

### Example: Solve |x+2| = 5

Using **"|u| = a is the same as u = ±a**":

Which has two solutions:

x+2 = −5 | x+2 = +5 |

x = −7 |
x = 3 |

## Graphically

Let us graph that example:

|x+2| = 5

It is easier to graph when we have an "=0" equation, so subtract 5 from both sides:

|x+2| − 5 = 0

Now let's plot **y=|x+2|−5** and find where it equals zero.

Here is the plot of y=|x+2|−5, but just for fun let's **make the graph by shifting it around**:

Start with y=|x| |
then shift it left to make it y=|x+2| |
then shift it down to make it y=|x+2|−5 |

And the two solutions (circled) are −7 and +3.

## Absolute Value Inequalities

Mixing Absolute Values and Inequalites needs a little care!

There are 4 inequalities:

< | ≤ | > | ≥ | |
---|---|---|---|---|

less than | less than or equal to |
greater than | greater than or equal to |

## Less Than, Less Than or Equal To

With "<" and "≤" we get **one interval** centered on zero:

### Example: Solve |x| < 3

This means the distance from **x** to zero must be less than 3:

Everything in between (but not including) −3 and 3

It can be rewritten as:

−3 < x < 3

As an interval it can be written as:

(−3, 3)

The same thing works for "Less Than or Equal To":

### Example: Solve |x| ≤ 3

Everything in between **and including** -3 and 3

It can be rewritten as:

−3 ≤ x ≤ 3

As an interval it can be written as:

[−3, 3]

How about a bigger example?

### Example: Solve |3x-6| ≤ 12

Rewrite it as:

−12 ≤ 3x−6 ≤ 12

Add 6:

−6 ≤ 3x ≤ 18

Lastly, multiply by (1/3). Because we are multiplying by a positive number, the inequalities will not change:

−2 ≤ x ≤ 6

**Done!**

As an interval it can be written as:

[−2, 6]

## Greater Than, Greater Than or Equal To

This is different ... we get **two separate intervals**:

### Example: Solve |x| > 3

It looks like this:

less than -3 **or** from 3 onwards

It can be rewritten as

x < −3 *or* x > 3

As an interval it can be written as:

(−∞, −3) U (3, +∞)

*Note: *U* means "Union" of the two intervals.*

Careful! **Do not** write it as

−3 > x > 3

Because **x** cannot be less than -3 **and** greater than 3 at the same time

It is really:

x < −3 *or* x > 3

**x** is less than −3 **or** greater than 3

The same thing works for "Greater Than or Equal To":

### Example: Solve |x| ≥ 3

Can be rewritten as

x ≤ −3 *or* x ≥ 3

As an interval it can be written as:

(−∞, −3] U [3, +∞)